Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #47589 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4075 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,3)(7,9)(8,10)(14,15)(17,18)(19,24)(20,26)(21,27)(22,30)(23,25)(28,31)(29,32)(33,34,39,44)(35,41,36,40)(37,42,47,45)(38,46,48,43);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19)(18,24)(20,29)(21,22)(23,28)(25,31)(26,32)(27,30)(33,35,39,36)(34,40,44,41)(37,46,47,43)(38,45,48,42);(5,13)(8,10)(11,16)(14,15)(18,25)(19,28)(22,26)(27,29)(33,35,39,36)(34,40,44,41)(37,43,47,46)(38,42,48,45);(1,3,6,2)(4,7,12,9)(5,10,13,8)(11,14,16,15)(17,20)(18,26)(19,29)(21,31)(22,25)(23,30)(24,32)(27,28)(33,36,39,35)(34,41,44,40)(37,43,47,46)(38,42,48,45);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21)(18,27)(19,26)(20,24)(22,28)(23,32)(25,29)(30,31)(33,37,39,47)(34,42,44,45)(35,43,36,46)(38,40,48,41);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,26)(18,20,25,30)(19,21,28,32)(24,29,31,27)(33,38)(34,43)(35,45)(36,42)(37,40)(39,48)(41,47)(44,46)Polycyclic presentation:
2^9|a^2=hi,b^2=g,c^2=hi,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8806088);